levenc/posologyr

Objective function: transpose of ETA

marianklose opened this issue · 1 comments

marianklose commented

Short question: For the calculation of the objective function we define:

U_eta <- eta %*% solve_omega %*% eta.

The formula from https://pubmed.ncbi.nlm.nih.gov/22563254/ however is proposing to take the transpose of eta one time. So we would end up with:

U_eta <- t(eta) %*% solve_omega %*% eta

Or am I missing something? I've found it in the definition of the objective function.

# Objective function for the Empirical Bayes Estimates
# doi: 10.4196/kjpp.2012.16.2.97
objective_function <- function(y_obs=NULL,f=NULL,g=NULL,
                               eta=NULL,solve_omega=NULL){

  # 1) When the prediction f is zero, g can be zero (depending on the residual
  # error model).
  # 2) log(0) is NaN, the limit of log(x) when x approaches zero is -Inf,
  # 3) log(1) is zero, and 1^2 is 1
  g[which(g == 0)] <- 1

  U_y   <-  sum(((y_obs - f)/g)^2 + log(g^2))

  #the transpose of a diagonal matrix is itself
  U_eta <- eta %*% solve_omega %*% eta

  if (TRUE %in% is.na(f)){
    # if rxode2 fails to solve the model, the proposed ETA is not optimal, assign
    # a large value to OFV to divert the algorithm from this area
    OFV <- 10^10
  } else {
    OFV <- U_y + U_eta
  }

  return(OFV)
}

Thank you!

levenc commented

Short question: For the calculation of the objective function we define:

U_eta <- eta %*% solve_omega %*% eta.

The formula from https://pubmed.ncbi.nlm.nih.gov/22563254/ however is proposing to take the transpose of eta one time. So we would end up with:

U_eta <- t(eta) %*% solve_omega %*% eta

That's correct. However here t(eta) and eta are equivalent. That is the meaning of the following comment :

#the transpose of a diagonal matrix is itself
  U_eta <- eta %*% solve_omega %*% eta

As eta is a diagonal matrix (it is unidimensional), its transpose is itself.
Hope this is helpful

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